Jack Preuveneers

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2026–Open question

Is the dispersion relation physics, or is it structure?

The swimming work produced something suspiciously tidy: eigenvectors that nearly separate into a spanwise shape times a chordwise wave, a spectrum sorted into branches by transverse order, growth rates arriving in matched pairs, and eigenvalues tracking a dispersion relation. The obvious reading is that this is fluid mechanics. The alternative is that it is linear algebra, and the fluid is incidental. This is the attempt to tell the two apart.

Python · NumPy / SciPy · LAPACK

The question

A dispersion relation showing up in a hydrodynamic problem invites a physical story: waves on a body, convection, wakes. But the discretised operators in that problem have a very particular shape, and shapes have consequences of their own.

If the wave organisation is physical, it tells you something about swimming. If it is structural, it tells you something far more general, and the hydrodynamics is just one instance of it. The two readings are not distinguishable by staring at the fluid results, so the test has to remove the fluid.

Two candidate explanations, fluid mechanics and matrix structure, both consistent with the same observed wave organisation.fluid mechanicswaves, wake, convectionmatrix structuretwo-level Toeplitz, decaywhat was observedseparable modes, branches,a dispersion relationBoth explain the evidence. Only one survives removing the physics.

The toy, with the physics taken out

The test object is a generalised eigenvalue problem built to have the same shape as the hydrodynamic one and none of its content. The first matrix is a random symmetric two-level Toeplitz matrix with independent normally distributed coefficients, optionally damped with distance. The second is a convection-type operator, a finite-difference first derivative plus a frequency term.

No physical quantity appears anywhere in it. Frequency, velocity and grid spacing are all set to one. If the organisation still appears, it cannot have come from the physics, because there is none.

The toy generalised eigenvalue problem: a dense random two-level Toeplitz matrix with decaying entries, set against a narrow convection-type difference operator.Arandom BTTB, decayingv = λBconvection typevFrequency, velocity and grid spacing are all set to one. No physics remains.

What two-level Toeplitz means

A Toeplitz matrix is constant along its diagonals: the entry depends only on the difference between its indices, not on where it sits. Block-Toeplitz-Toeplitz-block, or BTTB, applies that twice over. The matrix is made of blocks that repeat along block diagonals, and each block is itself Toeplitz.

That is the natural structure for a translation-invariant interaction discretised on a regular two-dimensional grid, which is exactly what a panel or lattice method on a rectangular planform produces. The second ingredient is decay: coefficients that fall off with distance, rather than staying the same size across the whole matrix.

A block-Toeplitz matrix whose blocks are themselves Toeplitz: the same block repeats along each block diagonal, and inside every block the same entry repeats along each diagonal.outer level: blocks repeat along block diagonalsEach block is itselfToeplitz: constantalong its diagonals.Entry depends only onthe index differences,at both levels.colour: magnitude, decaying with distance from the diagonal

The periodic case is exact

Wrap the structure onto a torus and it becomes block-circulant with circulant blocks, and then the answer is not approximate at all. Every two-dimensional Fourier mode is an exact eigenvector of both matrices simultaneously, and the eigenvalue is exactly the ratio of the two symbols evaluated at that wavenumber pair.

Verified to roundoff: a maximum pencil residual of 2.4 x 10⁻¹⁴, and agreement with a general dense solver to 6.2 x 10⁻¹⁵ relative. The correspondence between eigenpair and wavenumber is exact, structural and entirely free of physics.

The circulant case, where diagonals wrap around the corners of the matrix, making every Fourier mode an exact eigenvector of both matrices.diagonals wrap at the cornersEvery 2-D Fourier mode isan exact eigenvector ofboth matrices at once.residual 2.4 x 10⁻¹⁴agreement 6.2 x 10⁻¹⁵Exact, structural, and with no physics anywhere in it.

The finite case, which is the real one

Real problems are not periodic, and the finite case is where it could have fallen apart. It mostly does not. The transverse reduction is exact for the partially wrapped pencil, and the finite pencil departs from it only by a measured wrap error. The spectrum decouples into exactly as many branches as there are transverse orders, with the label readable from the eigenvector.

Eigenvalues track the predicted symbol ratio with a median error of 4.1% when the coefficients decay, against 58% when they do not. The spectrum is exactly symmetric under a conjugate reflection, so envelope rates arrive in matched positive and negative pairs. For the symmetric toy those rates are measurably zero, and making the coefficients asymmetric along the chord switches on genuine growth and decay.

The finite spectrum separating into four branches, each labelled by its transverse standing order, with the label readable from the eigenvector rather than the eigenvalue.n = 1n = 2n = 3n = 4exactly as many branches as transverse ordersEigenvalues track the predicted ratio to 4.1% with decay, 58% without.

The null experiment

The sharpest evidence is a comparison against matrices that have only some of the ingredients. Separability of the eigenvectors is 0.999 for random two-level Toeplitz with decaying coefficients, 0.380 for the same structure without decay, and 0.255 for a random symmetric matrix with no Toeplitz structure at all.

Neither ingredient is sufficient alone, and together they are enough in practice. Sweeping the decay exponent locates the breakdown near the point where the coefficients stop being summable over the lattice, which is the threshold the theory would predict. Changing the random seed changes individual eigenpairs and leaves every statistic, the branch census and the model subspaces intact: the organisation is a property of the structure, not of the draw.

0.999
separability, BTTB with decay
0.380
BTTB without decay
0.255
no Toeplitz structure
Median eigenvector separability: 0.999 for a random two-level Toeplitz matrix with decaying coefficients, 0.380 without decay, and 0.255 for a random symmetric matrix with no Toeplitz structure.1.000.999BTTBwith decay0.380BTTBno decay0.255randomsymmetricMedian SVD separability, 20x20, random coefficients. Both ingredients are needed.
Separability rising with the decay exponent, from 0.43 at one half to 0.99 at three, crossing the lattice summability threshold at an exponent of two.summability threshold0.430.50.510.8220.9931.0Separability against radial decay exponent.

What an eigenvalue cannot tell you

A tempting shortcut is to read the wavenumber straight off the eigenvalue. That does not work, and the reason is exact rather than numerical: equal and opposite transverse wavenumbers collide onto the same eigenvalue, to 9 x 10⁻¹⁶. Branch regions also interpenetrate in the complex plane, so identifying a branch from the eigenvalue alone performs at chance.

Given the branch label, which the eigenvector does carry, the level-set equation inverts branch by branch. A small Newton solve converges every time and recovers the chordwise wavenumber to a median of 0.036 radians, about 11% of the grid spacing. So the modes are wavenumber-labelled, but conditionally, and the eigenvector is doing work the eigenvalue cannot.

Two different transverse wavenumbers, plus and minus theta, mapping to the same eigenvalue, so the eigenvalue alone cannot identify the mode.k_y = +θk_y = −θone λIdentical to9 x 10⁻¹⁶.Branch identification from λ alone performs at chance.The eigenvector carries the label the eigenvalue cannot.

What it would mean, and what is still open

If the organisation is structural, then the classification found in the swimming problem is not a fact about swimming. It should appear in any translation-invariant interaction on a regular grid paired against a convection-like operator, and the physics would enter only through the numerical values of the coefficients, not through the shape of the answer. That makes the method portable, and it also means a physical interpretation of the branches has to be argued for rather than assumed.

The honest gaps: the classical theory for banded Toeplitz operators does not cover a dense-but-decaying pencil, so part of the finite-size argument rests on numerical evidence with the obstruction stated rather than on proof. The study is tagged throughout by rigour, separating what is exact for the periodic case, exact for the finite case, asymptotic, numerical, and still conjecture. Those tags are the point: the claim is only as strong as the weakest link, and the weakest link is named.

The five rigour tags used throughout the study, from exact and verified at the top down to conjecture with the obstruction named at the bottom.every claim carries its own standard of proofEXACT-PERIODICproved, verified to roundoffEXACT-FINITEproved for reduction and symmetryASYMPTOTICstandard theory, stated conditionsNUMERICALmeasured, seed invariantCONJECTUREobstruction named, not closedThe claim is only as strong as its weakest link, so the weakest link is named.